Please see the attached file for the fully formatted problems.
Let (Omega, A) be a measurable space, and P:A--> [0,infinity] an application such that P(AUB) = P(A) + P(B) when A,B E A and A intersection B = ø, and P(Omega) = 1 . Prove that the following statements are equivalent:
(i) P is a probability
(ii) P is continuous on the decreasing series:
(iii) P is continuous on the increasing series:
(iv) and ø
Proofs are provided for proabilities expressed with set theory. The solution is detailed and well presented.